IEEE Circuits and Systems Magazine - Q3 2022 - 9

Dnh = h
df ()
Z
[
\
]]
]
1
,
,
n
n
! 0
=
.
The anti-causal unit step is given by ().nhf -
we obtain
Dnh
T ()f -= h
Z
[
\
]]
]
-=,
1
,
n
n
! 0
.
(6)
Using (2)
(7)
The above defined derivatives enjoy the same properties
as the continuous-time (CT) derivatives [22], namely
■ Linearity
■ Causality
■ Time reversal
The substitution
tt ,&■
Additivity and Commutativity of the orders.
Let a and b be real numbers. Then,
DD ft DD ft Df t==ab
66 ()
()@@ ()
ba
ab+
■ Neutral element
The existence of neutral element is a consequence of
the previous property. Letting
DD ft Df tf t
aa ==
-
6
()@
0 () ().
■ Inverse element
From the last result we conclude that there is always
an anti-derivative.
Example II.2. Fractional derivatives of impulses
It is straightforward to show that the derivative of any order
of the impulse is essentially given by the binomial coefficients.
In fact, from (3) and (4) we get
Dn h
d
a
and
Dn =-h
T
a
d () ()
--a 1 -a -n
-n
()!
()
f().
-nh
(9)
According to the above properties, it is easy to obtain
the fractional derivative of the step functions. We only
have to substitute
Dnhhf()=
d
a
and
and
Dnhhf() ()
T
a
=THIRD
QUARTER 2022
-a -+1 -n
()!
()
-n
a
f().
-nh
De (, )(,).
T TT
a
ts se ts
a
=
(16)
IEEE CIRCUITS AND SYSTEMS MAGAZINE
9
a 1- for a and divide by h, so that:
()
-a -+1 n
n!
a
f nh
()
et se ts et sTTd
=-1 =- -
(, )/ (, )( ,).
2) As h 0 " both exponentials converge to .est
3) Eigenfunctions
De (, )(,)
d dd
a
ts se ts
a
=
(15)
(14)
d ()=
--a
1 -a n
n!
()
f nh
()
(8)
Example II.3. Fractional derivatives of the discrete
power functions
For negative values of a these expressions can be considered
the definitions of fractional discrete " powers " . Their
derivatives are given by
D ;
d
b
()
!
n
a
nn1
f
()Enh = h-+b
()
a - b
n!
f nh
(),
(10)
which represents the analogue of the derivative of the power
function. Similarly we obtain for the delta derivative
D ;
T
b
()!
()
a -
-n
nn1
f
() ()-+
-= -h
nh E
b
()!
()
a - b
-n
f().
-nh
(11)
Remark II.1. It can be shown that these " powers " tend
converts the forward
(nabla) derivative into the backward (delta) and
vice-versa.
to the causal CT powers when h 0 " [27].
Remark II.2. The theory described in this and in the
following sub-sections can be generalised for irregular
time scales. However, such study goes beyond the
objectives of this work [27].
B. The Nabla and Delta Exponentials
As we discussed for the CT LS [2], the usual (eternal)
exponentials,
e ,st
ba ,=- then it results
,
z ,n n Z! , are the eigenfunctions of
t R! and s C! are the eigenfunctions
of such systems. As it is well known, the discretetime
exponentials,
the discrete systems described by difference equations.
Here, we introduce the exponentials suitable for dealing
with the systems defined by the nabla and delta derivatives.
These exponentials play a similar role to those
discussed in the study of CT LS.
The nabla exponential is defined by:
etssh1
d =- (, ),
6
where
.
etssh1
T =+6
@-th/
@
th/
(12)
s C! Similarly, the delta exponential is given by:
(, ).
11 and 6 +=@
sh
sh
(13)
The corresponding complex sinusoids are obtained
when s is over the circles 6 -=@
11 ,
s C! , respectively. They are called right and left Hilger
circles [15], [16]. The main properties of the exponentials
read [22]
1) Relation between the nabla and delta exponentials:
IEEE Circuits and Systems Magazine - Q3 2022

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q3 2022
